Quantum effects in nanosystems: Good reasons to use phase-space Weyl symbols

Bogoliubov transformations have been successfully applied in several condensed-matter contexts, e.g., in the theory of superconductors, superfluids, and antiferromagnets. These applications are based on bulk models where translation symmetry can be assumed, so that few degrees of freedom in Fourier space can be “diagonalized” separately, and in this way it is easy to find the approximate ground state and its excitations. As translation symmetry cannot be invoked when it comes to nanoscopic systems, the corresponding multidimensional Bogoliubov transformations are more complicated. For bosonic systems it is much simpler to proceed using phase-space variables, i.e., coordinates and momenta. Interactions can be accounted for by the self-consistent harmonic approximation, which is naturally developed using phase-space Weyl symbols. The spin-flop transition in a short antiferromagnetic chain is illustrated as an example. This approach, rarely used in the past, is expected to be generally useful to estimate quantum effects, e.g., on phase diagrams of ordered vs disordered phases.

Figure 1

Minimum-energy configuration of (the classical counterpart of) the AFM described by (44): for small field, the spins of the two sublattices are antiparallel along the $z$ axis; beyond a given field $H_{{}_{\mathrm{SF}}}$ the spin-flop configuration prevails, with the angle $\mathrm{\Theta }$ increasing with $H$ up to $\pi /2$ at the saturation field.

Figure 2

Phase diagram for the finite chain with an odd number of classical spins $N$. The curves report the analytical result derived in Appendix pp1, which interpolates the points at odd values of $N$. For easy-plane anisotropy $\mu <1$ the AP configuration exists between two field values $h_{\mp }$ (those for $\mu =0.999$ and $N=13$ are shown), and there is a maximum $N$ beyond which only the SF phase exists; for $\mu \ge 1$ the AP phase persists for $h\to 0$, i.e., $h_{-}=0$, and the AP phase exists for any $N$.